Due to these caveats, it remains to be explored how to associate Chern numbers to discrete chains such as the Thue-Morse and Period-Doubling chains, where a relation to topological boundary effects and pumping was not established. Considering cut-and-project quasicrystals, the interpolating potential (REF ) is equivalent to “smoothening” the cut's acceptance window, whereas we study the response system to a phason shift of the window, cf. Fig. REF (b) and Ref. [1]}. Such a smoothening in higher-dimensional quasicrystals is challenging, e.g., in the Amman-Beekner tiling the cut-window is octagonal, and can be modified in different ways. Additionally, in the presence of several quasiperiodic potentials, the length scales competition can become more elaborate than discussed here, and the gaps can close as a function \(\phi \)  and \(\theta \) .
